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Displaying similar documents to “The primitive Boolean matrices with the second largest scrambling index by Boolean rank”

Linear operators that preserve graphical properties of matrices: isolation numbers

LeRoy B. Beasley, Seok-Zun Song, Young Bae Jun (2014)

Czechoslovak Mathematical Journal

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Let A be a Boolean { 0 , 1 } matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A . A linear operator on the set of m × n Boolean matrices is a mapping which is additive and maps the zero matrix, O , to itself. A mapping strongly preserves a set, S , if it maps the set S into the set S and the complement...

Linear operators that preserve Boolean rank of Boolean matrices

LeRoy B. Beasley, Seok-Zun Song (2013)

Czechoslovak Mathematical Journal

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The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m × k Boolean matrix B and a k × n Boolean matrix C such that A = B C . In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2 . In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves...

On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall, Kyriakos Keremedis (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product 2 ( ω ) the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of 2 ( ω ) to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean...